#108317

order	512
nilpotency class	3
isoclinism	isoclinic class of #108316
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2)(3,7,4,6)(5,8)(9,10)(11,21,24,34)(12,25,28,32)(13,18)(14,19)(15,20)(16,23,31,29)(17,27,33,26)(22,30)(35,38)(36,39)(37,40);(1,3,5,4)(2,6,8,7)(9,11)(10,16,20,31)(12,22)(13,23)(14,29)(15,24)(17,30,33,40)(18,21,19,34)(25,35,32,39)(26,36)(27,38)(28,37);(1,4,5,3)(2,7,8,6)(9,12)(10,17)(11,22)(13,27)(14,26)(15,28)(16,30)(18,25)(19,32)(20,33)(21,35)(23,38)(24,37)(29,36)(31,40)(34,39)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,e^b=ei,f^b=fhi,d^c=di,e^c=ei,f^c=fhi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15513);